andrew_gelman_stats andrew_gelman_stats-2010 andrew_gelman_stats-2010-355 knowledge-graph by maker-knowledge-mining
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Introduction: Last week, as I walked into Andrew’s office for a meeting, he was formulating some misgivings about applying an ideal-point model to budgetary bills in the U.S. Senate. Andrew didn’t like that the model of a senator’s position was an indifference point rather than at their optimal point, and that the effect of moving away from a position was automatically modeled as increasing in one direction and decreasing in the other. Executive Summary The monotonicity of inverse logit entails that the expected vote for a bill among any fixed collection of senators’ ideal points is monotonically increasing (or decreasing) with the bill’s position, with direction determined by the outcome coding. The Ideal-Point Model The ideal-point model’s easy to write down, but hard to reason about because of all the polarity shifting going on. To recapitulate from Gelman and Hill’s Regression book (p. 317), using the U.S. Senate instead of the Supreme Court, and ignoring the dis
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1 Andrew didn’t like that the model of a senator’s position was an indifference point rather than at their optimal point, and that the effect of moving away from a position was automatically modeled as increasing in one direction and decreasing in the other. [sent-4, score-0.747]
2 Executive Summary The monotonicity of inverse logit entails that the expected vote for a bill among any fixed collection of senators’ ideal points is monotonically increasing (or decreasing) with the bill’s position, with direction determined by the outcome coding. [sent-5, score-1.242]
3 y[i] is the vote by senator j[i] on bill k[i]. [sent-11, score-1.087]
4 The value alpha[j] is the ideal point of senator j and the value beta[k] the position of bill k. [sent-12, score-1.318]
5 ” To identify the parameters, we can center the alphas around 0 and assume that liberal senators will have smaller alpha[j] and conservative senators larger alpha[j] values. [sent-16, score-0.908]
6 On the same scale, more conservative bills will have larger beta[k] and more liberal bills smaller beta[k]. [sent-17, score-0.792]
7 Given a vote on bill k with position beta[k], the further the senator’s position is to the right (i. [sent-21, score-1.148]
8 It’s colored red because a conservative bill is coded with a vote of y[i]=1 being for the bill. [sent-44, score-1.043]
9 Thus the red curve is the number of votes for a conservative bill based on its position. [sent-45, score-0.879]
10 The blue curve shows the number of votes for a liberal bill based on its position. [sent-49, score-0.911]
11 Thus the coding of a bill makes a difference to the position at which it’ll pass as well as the direction of movement which increases likelihood of passage. [sent-50, score-0.887]
12 Specifically, for a bill coded as conservative, the more liberal its position, the more likeliy it is to pass. [sent-51, score-0.813]
13 Making a bill more liberal or more conservative has the same effect on senators of both parties, making them either more or less likely to vote for the bill, depending on how the coding is chosen. [sent-54, score-1.43]
14 Flipping the coding actually changes the point at which a bill will pass, and seems redundant given that a bill’s position, beta[k], is also coding a bill’s liberal/conservative orientation. [sent-55, score-0.669]
15 A More Concrete Example For concreteness, consider the US$ 800+ 2009 stimulus bill , a liberal bill which squeaked through the U. [sent-56, score-1.298]
16 If the bill moves to the left, beta[k] decreases, so alpha[i]-beta[k] inreases, and the vote is more likely to be 1 (against the stimulus). [sent-64, score-0.835]
17 Thus no matter what the senator’s ideal point is, they’re more likely to vote against the bill as the target amount becomes more liberal (rises). [sent-65, score-1.246]
18 If the bill moves to the right, presumably corresponding to a smaller stimulus package, beta[k] increases, so alpha[i]-beta[k] decreases, so the vote is more likely to be 0 (for the stimulus). [sent-66, score-1.166]
19 Then, the further the bill got away from the senator’s ideal in either direction, the more likely they’d be to vote against it, rather than assuming their preference for the bill goes up in one direction of change and down in the other relative to their indifference point. [sent-71, score-1.556]
20 A hypothetical bill with a position to the left (or right) of all the senators will still act monotonically, but now with the expected directional effect, namely increasing expected votes as the bill approaches the senator’s ideal points. [sent-72, score-2.038]
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